In my talk I would like to discuss the present status of Doubly Special Relativity. DSR is an extension of Special Relativity aimed at describing kinematics of particles and fields in the regime where (quantum) gravity effects might become relevant. I will discuss an interplay between DSR physics and mathematics of Hopf algebras.

A brief introduction to the notorious "cosmological constant problem" is given. Then, a particular approach is discussed, which has been developed by Volovik and the present speaker over the last years and which goes under the name of q-theory. This approach provides a possible solution of the main cosmological constant problem, why is |Lambda|^(1/4) negligible compared to the energy scales of the electroweak standard model (not to mention the Planck energy)?

Combining the principles of general relativity and quantum theory still remains as elusive as ever. Recent work, that concentrated on one of the points of contact (and conflict) between quantum theory and general relativity, suggests a new perspective on gravity. It appears that the gravitational dynamics in a wide class of theories - including, but not limited to, standard Einstein's theory - can be given a purely thermodynamic interpretation. In this approach gravity appears as an emergent phenomenon, like e.g., gas or fluid dynamics.

Attempts to go beyond the framework of local quantum field theory include scenarios in which the action of external symmetries on the quantum fields Hilbert space is deformed. A common feature of these models is that the quantum group symmetry of their Hilbert spaces induces additional structure in the multiparticle states which in turns reflects a non-trivial momentum-dependent statistics.

Loop quantum gravity and spin foams are two closely related theories of quantum gravity. There is an expectation that the sum over histories or path integral formulation of LQG will take the form of a spin foam, although a rigorous connection between the two is available only in 2+1 gravity. Understanding the relation between them will resolve many open questions of both theories. We probe the connection through an exactly soluble model of loop quantum cosmology. Beginning from the canonical theory we construct a spin foam like expansion of LQC.

I will describe a very special (infinite-parameter) family of gravity theories that all describe, exactly like General Relativity, just two propagating degrees of freedom. The theories are obtained by generalizing Plebanski's self-dual (chiral) formulation of GR. I will argue that this class of gravity theories provides a potentially powerful new framework for testing the asymptotic safety conjecture in quantum gravity.

A quantum theory of gravity implies a quantum theory of geometries. To
this end we will introduce different phases spaces and choices for the
space of discretized geometries. These are derived through a canonical
analysis of simplicity constraints - which are central for spin foam
models - and gluing constraints. We will discuss implications for
spin foam models and map out how to obtain a path integral
quantization starting from a canonical quantization.

Diffeomorphism symmetry is the underlying symmetry of general
relativity and deeply intertwined with its dynamics. The notion of
diffeomorphism symmetry is however obscured in discrete gravity, which
underlies most of the current quantum gravity models. We will propose
a notion of diffeomorphism symmetry in discrete models and find that
such a symmetry is weakly broken in many models. This is connected to
the problem of finding a consistent canonical dynamics for discrete
gravity. Finally we will discuss methods to construct models with

We will give a short overview of non-perturbative quantum gravity
models and discuss some key common problems for these models. In
particular we will analyze what background independence requires from
a theory of quantum gravity.

In causal set quantum gravity, spacetime is assumed to have a fundamental atomicity or discreteness, and is replaced by a locally finite poset, the causal set. After giving a brief review of causal sets, I will discuss two distinct approaches to constructing a quantum dynamics for causal sets. In the first approach one borrows heavily from the continuum to construct a partition function for causal sets.